Nabla (∇)

How to read it

The name nabla comes from a harp. William Robertson Smith suggested it to P. G. Tait because the inverted delta looked like an Assyrian harp, according to Cargill Knott’s 1911 life of Tait, quoted in Jeff Miller’s Earliest Known Uses of Some of the Words of Mathematics. Tait and James Clerk Maxwell used the name playfully, and Maxwell’s one published use of it was in the title of his humorous “Tyndallic Ode”, dedicated to the “Chief Musician upon Nabla”, Tait. The other name, del, is in print by 1901, in Edwin Bidwell Wilson’s Vector Analysis. William Rowan Hamilton introduced the operator, writing it on its side, and Tait established ∇ as the standard symbol in 1867 (Miller).

Also searched as: ∇, del operator, gradient symbol, nabla symbol, upside down triangle symbol.

The nabla character and look-alikes

Unicode’s chart gives nabla the aliases backward difference, gradient and del, and notes it is used for the Laplacian operator, written with a superscript 2. Unicode cross-refers it to the white down-pointing triangle ▽, a separate character. The right-way-up triangle is the increment sign ∆ or the Greek capital Delta Δ.

Nabla in maths

The gradient ∇f is the vector of a function’s partial derivatives; OpenStax’s Calculus Volume 3 says “the symbol ∇ is called nabla and the vector ∇f is read ‘del f’”. Written with a dot, ∇ · F is the divergence of a vector field F. Written twice, as ∇², it is the Laplace operator, and Laplace’s equation is ∇²f = 0 (OpenStax). Robert Murphy wrote the Laplacian as Δ in 1833 (Miller). In the calculus of finite differences ∇ has a separate meaning, the backward difference ∇f(x) = f(x) − f(x − 1) (NIST Digital Library of Mathematical Functions).

In AI and machine learning

Training a neural network by gradient descent moves the parameters against the gradient ∇. PyTorch’s optimiser documentation writes the gradient of the objective with respect to the parameters θ as ∇_θ f(θ), and Michael Nielsen’s Neural Networks and Deep Learning writes each step as Δv = −η∇C, the learning rate η times the gradient of the cost C.

In quantum physics

∇² appears in the Schrödinger equation for a particle in three dimensions. Lecture notes from the University of North Carolina Wilmington write the time-independent equation as −(ħ²/2m)∇²ψ + Vψ = Eψ, where ψ is the particle’s wave function.